Averaging in thermodynamic dislocation theory: general macroscopically uniform stress and strain states

Abstract

The averaging procedure, developed for polycrystalline bars under axially symmetric tension or compression, is extended to arbitrary macroscopically uniform stress and strain states. Starting from the equal probability hypothesis for grain orientations, the mean resolved shear stress and the mean resolved elastic and plastic shear strains are defined as root-mean-square averages over all slip-system orientations. Two exact identities for isotropic orientation averages show that the mean resolved shear stress is proportional to the von Mises equivalent stress, and that the direction of macroscopic plastic flow, obtained from the hypothesis that the plastic slip rate on a system is proportional to the resolved shear stress acting on it, is the stress deviator. The result is an associated $J_2$ flow theory whose hardening law is not fitted but follows from the kinetics of thermally activated dislocation depinning and the evolution equations for the dislocation density and the effective disorder temperature of thermodynamic dislocation theory, written as rates with respect to time so that arbitrary loading paths can be followed. For torsion the theory yields the torque--twist relation of bars and tubes without the classical reductions to a shear stress--strain curve. The parameters for copper are identified jointly from Hopkinson-bar tension, dynamic compression at room and elevated temperatures, and torque--twist records at several twist rates. One set of material parameters, consistent with the earlier compression-only identification, describes tension, compression and torsion from room temperature to $1173$\,K and from $10$ to $2300$\,s$^{-1}$ to $6$\,\% rms over 142 data points; the tension--torsion discrepancy noted by Johnson and Cook is traced to the initial dislocation state of their torsion specimens.